Add Folders to Path

We start by adding the necessary folders to the current working path.

Overview

In the following notebok we visualize the FTLE from AVISO data. The notebook is structured as follows:

  1. Import data from the file 'AVISO.mat' stored in the folder 'data'.
  2. Define computational parameters (such as the number of cores) and data.
  3. Define spatio-temporal domain.
  4. Interpolate velocity from (discrete) gridded data.
  5. Finite Time Lyapunov Exponent (FTLE):

    • Compute gradient of flow map $ \mathbf{\nabla F}_{t_0}^{t_N}(\mathbf{x}_0) $ over meshgrid using an auxiliary grid.

    • Compute Cauchy Green (CG) strain tensor as:

      \begin{equation} \mathbf{C}_{t_0}^{t_N}(\mathbf{x}_0) = \mathbf{\nabla F}_{t_0}^{t_N}(\mathbf{x}_0)^T\mathbf{\nabla F}_{t_0}^{t_N}(\mathbf{x}_0). \end{equation}

    • Compute maximum eigenvalues $ \lambda_2(\mathbf{x}_0) $ of $ \mathbf{C}_{t_0}^{t_N}(\mathbf{x}_0) $.

    • Compute FTLE as:

      \begin{equation} \mathrm{FTLE}_{t_0}^{t_N}(\mathbf{x}_0) = \dfrac{1}{|t_N-t_0|} \log(\sqrt{\lambda_2(\mathbf{x}_0)}). \end{equation}

Import Data

Computational parameters and data

Here we define the computational parameters and the data.

Spatio-temporal domain

Here we define the spatio-temporal domain over which to consider the dynamical system.

Velocity Interpolation

In order to evaluate the velocity field at arbitrary locations and times, we interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear".

Finite-Time Lyapunov Exponent (FTLE)

Next, we compute the FTLE over the meshgrid over the given time-interval. We iterate over all initial conditions and first calculate the gradient of the flow map using an auxiliary grid. 'aux_grid' specifies the ratio between the auxiliary grid and the original meshgrid. This parameter is generally chosen to be between $ [\dfrac{1}{5}, \dfrac{1}{10}] $. Subsequently, we compute the Cauchy Green strain tensor. From the maximum eigenvalue of the Cauchy-Green strain tensor we can then compute the FTLE. The iteration over the meshgrid is parallelized.

The forward FTLE field is plotted with respect to the initial position of the particles. Repelling LCS are visible as ridges in the forward FTLE field. An analogous statement holds for attracting LCS and the backward FTLE field. As a first approach, ridges of the FTLE field can be extracted using simple thresholding. This should already provide a good hint to the location of hyperbolic features.

FTLE ridges are, however, necessary but not sufficient indicators for hyperbolic LCS. In fact, ridges of the FTLE field have also been related to regions of high shear. As a consequence the FTLE should only be considered as a first (and relatively rapid) approach for finding hyperbolic (repelling/attracting) LCS. More advanced methods such as the computation of the tensorlines associated to the eigenvalues of the Cauchy-Green strain tensor, should be used for a precise computation of hyperbolic LCS (see HyperbolicLCS) [1, 2].

References

[1] Haller, G. (2015). Lagrangian coherent structures. Annual Review of Fluid Mechanics, 47, 137-162.

[2] Haller, G., & Sapsis, T. (2011). Lagrangian coherent structures and the smallest finite-time Lyapunov exponent. Chaos: An Interdisciplinary Journal of Nonlinear Science, 21(2), 023115.